Vibration isolation is the engineering practice of reducing the transfer of oscillatory motion or force between a machine and its support. This cheat sheet helps students connect spring, mass, and damper models to practical isolation design. It is useful for analyzing rotating equipment, precision instruments, vehicle mounts, and building mechanical systems.
The main goal is to choose stiffness and damping so harmful vibration is reduced over the required frequency range.
The core model is a single-degree-of-freedom mass-spring-damper system driven by either a harmonic force or moving base. The most important quantities are natural frequency, damping ratio, frequency ratio, transmissibility, and isolation efficiency. Isolation usually begins when the excitation frequency is greater than about sqrt(2) times the natural frequency.
Damping reduces resonance peaks but can reduce high-frequency isolation performance.
Key Facts
- The undamped natural circular frequency is omega_n = sqrt(k/m), where k is stiffness and m is supported mass.
- The natural frequency in hertz is f_n = (1 / 2 pi) sqrt(k/m).
- The damping ratio is zeta = c / c_c, where c_c = 2 sqrt(km) is the critical damping coefficient.
- The frequency ratio is r = omega / omega_n = f / f_n, where omega or f is the excitation frequency.
- For harmonic force isolation, force transmissibility is T = sqrt(1 + (2 zeta r)^2) / sqrt((1 - r^2)^2 + (2 zeta r)^2).
- Isolation occurs when T < 1, which for light damping begins approximately when r > sqrt(2).
- Isolation efficiency is eta = (1 - T) x 100 percent when T is less than 1.
- Static deflection under weight is delta = mg/k, and the vertical natural frequency can be estimated by f_n = (1 / 2 pi) sqrt(g/delta).
Vocabulary
- Transmissibility
- Transmissibility is the ratio of transmitted vibration force or motion to the input force or motion.
- Natural Frequency
- Natural frequency is the frequency at which a system tends to vibrate freely after being disturbed.
- Damping Ratio
- Damping ratio is a nondimensional measure of damping compared with critical damping.
- Frequency Ratio
- Frequency ratio is the excitation frequency divided by the system natural frequency.
- Static Deflection
- Static deflection is the steady displacement of an isolator caused by the supported weight.
- Resonance
- Resonance is the condition in which excitation frequency is near natural frequency and vibration amplitude can become large.
Common Mistakes to Avoid
- Using hertz and radians per second interchangeably is wrong because omega = 2 pi f, so missing the 2 pi factor changes natural frequency and transmissibility.
- Assuming more damping always improves isolation is wrong because damping lowers the resonance peak but increases transmitted motion at high frequency ratios.
- Designing for r near 1 is wrong because this places the system near resonance, where amplitudes and transmitted forces can be much larger than the input.
- Ignoring static deflection is wrong because a low natural frequency usually requires a soft isolator, which may sag too much or exceed clearance limits.
- Applying the force transmissibility formula to every base-motion problem is wrong because base excitation can require displacement transmissibility and relative motion checks.
Practice Questions
- 1 A 120 kg machine is mounted on isolators with total stiffness k = 48,000 N/m. Find omega_n and f_n.
- 2 An isolator has f_n = 6 Hz and the machine excitation frequency is 24 Hz. Find the frequency ratio r and state whether isolation is expected.
- 3 For zeta = 0.10 and r = 3, calculate the force transmissibility T using T = sqrt(1 + (2 zeta r)^2) / sqrt((1 - r^2)^2 + (2 zeta r)^2), then find isolation efficiency.
- 4 Explain why an isolator with a very soft spring can reduce transmitted vibration but still be unacceptable in a real machine installation.